Is the VC dimension of an origin-centered circle 1 or 2?
It seems to me that the VC dimension of an origin centered circle should be 1, because for two points with distances from the origin r1 <= r2, r2 will never be able to be labeled 1 without r1 also being labeled 1, so the r1 = 0, r2 = 1 labeling could never be possible.
However, this and this both say that an origin-centered circle has VC dimension of 2.
What you are missing is a definition of origin-centered circles. The definition of the slides (your second link) is wrong, for the reasons you mention. The definition in the lecture notes (your first link) makes it clear that you can choose whether it is the inside or the outside of the circle which gets the value +1, and this shows that any two points whose distance from the origin is not the same can be shattered, and so the VC dimension is at least 2.
Assume that the points on the circle are considered inside. With that if you have 2 points at an equal distance from the origin, you cannot label one + and the other -. Either both are in or both are out. On the contrary for 3 points with varying distance from the origin, you can label them +, -, +. This arrangement cannot be shattered. Hence the VC dimension is 2.
Consider the distance as an attribute and +/- as a label. For the same set of attributes, you cannot have a different label.
Why are you excluding the case where the distance is the same? Take any two points with equal distanced from the origin and they cannot be shatters -> VC is 1
